1. Let (xn)n be the sequence given by xn = 1/ n^ 2 + sin(n/6). This sequence...

1.

Let (xn)n be the sequence given by xn = 1/ n^ 2 + sin(nπ/6). This sequence has exactly 7 distinct subsequential limits.

a. Find the subsequential limits by finding 7 subsequences (be precise on what the corresponding nk should be), each converging to one of the subsequential limits. (No proof necessary.)

b. Prove (using Theorem 52: A number z is a subsequential limit of a sequence(xn)n

if and only if for everyε >0, the set of indices{n∈N:|xn−z|< ε} is infinite.)

that no other real number can be a subsequential limit